— title: “Section2_Lab03_SwaminathanV_PalitS” output: html_document —
Abstract
Big box retailers such as Walmart need to have accurate models of purchasing patterns at their various stores. Overestimating consumer demand could lead to losses due to excess inventory.
On the other hand, underestimating it could result in lower customer service level and lost sales.
Our dataset contains historical sales data for 45 Walmart stores located in different regions. Each store contains a number of departments.
For the purpose of this lab, we are analyzing a randomly chosen store 4 and department 12 ( based on a random number generator)
Is there a relationship between temperature and sales?
Is there a relationship between CPI and sales?
Is there a relationship between fuel price and sales?
Is there a relationship between unemployment and sales?
Finally, we are going to forecast sales for this store which will help them maximize sales and optimize inventory.
Loading the source datasets and the required libraries
train<-read.csv('/Users/vyas/Dropbox/Berkeley MIDS/Advanced Regression/W271 Lab 3/train.csv')
features<-read.csv('/Users/vyas/Dropbox/Berkeley MIDS/Advanced Regression/W271 Lab 3/features.csv')
train <- train[train['Dept']==12,]
train<-train[,c(1,2,3,4)] Function to perform EDA on time series
tseda<-function(timeseries,xlabel,ylabel,mainlabel){
plot(timeseries,xlab=xlabel,ylab=ylabel,main=mainlabel)
#par(mfrow=c(1,2))
Acf(timeseries,main = paste(mainlabel,' ACF'))
Pacf(timeseries,main =
paste(mainlabel,' Pacf'))
adf.test(timeseries)
}Function to perform out of sample tests on forecasts using RMSE
rmse<-function(timeseries1,timeseries2){
return(sqrt(mean((timeseries1-timeseries2)^2))) }Merging and filtering the data
input<-merge(train,features,by =c("Store","Date"))
input<-input[,c(1,2,3,4,5,6,12,13)]
input<-input[(input["Store"]==4)|(input["Store"]==14)|(input["Store"]==15),]
store4_data<-input[input["Store"]==4,]Describe the time series in the dataset
str(store4_data) ## 'data.frame': 143 obs. of 8 variables:
## $ Store : int 4 4 4 4 4 4 4 4 4 4 ...
## $ Date : Factor w/ 143 levels "2010-02-05","2010-02-12",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ Dept : int 12 12 12 12 12 12 12 12 12 12 ...
## $ Weekly_Sales: num 8245 6689 6791 8821 8800 ...
## $ Temperature : num 43.8 28.8 36.5 41.4 43.5 ...
## $ Fuel_Price : num 2.6 2.57 2.54 2.59 2.65 ...
## $ CPI : num 126 126 127 127 127 ...
## $ Unemployment: num 8.62 8.62 8.62 8.62 8.62 ...summary(store4_data)## Store Date Dept Weekly_Sales Temperature
## Min. :4 2010-02-05: 1 Min. :12 Min. : 4209 Min. :28.84
## 1st Qu.:4 2010-02-12: 1 1st Qu.:12 1st Qu.: 7277 1st Qu.:48.47
## Median :4 2010-02-19: 1 Median :12 Median : 8336 Median :64.22
## Mean :4 2010-02-26: 1 Mean :12 Mean : 8142 Mean :62.25
## 3rd Qu.:4 2010-03-05: 1 3rd Qu.:12 3rd Qu.: 9037 3rd Qu.:77.44
## Max. :4 2010-03-12: 1 Max. :12 Max. :10870 Max. :86.09
## (Other) :137
## Fuel_Price CPI Unemployment
## Min. :2.540 Min. :126.1 Min. :3.879
## 1st Qu.:2.764 1st Qu.:126.6 1st Qu.:4.607
## Median :3.290 Median :129.1 Median :5.946
## Mean :3.217 Mean :128.7 Mean :5.965
## 3rd Qu.:3.587 3rd Qu.:130.5 3rd Qu.:7.127
## Max. :3.881 Max. :131.2 Max. :8.623
## Based on an analysis of the structure of the data, it doesn’t appear as though there are missing values in the data set.
Examine the various series for stationarity
- The ACF and PACF plots for all the time series show the classic signature for an AR model of root 1. The ACF drops down slowly while the PACF drops suddenly at lag 1. This series’ will need to be differenced at least once to get to stationarity
Time Series 1: Sales
store4_sales.ts<-ts(store4_data$Weekly_Sales,start =c(2010,5),freq=52)
tseda(store4_sales.ts,"Year","Sales at Store 4","Store 4
Sales by Year") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -3.0062, Lag order = 5, p-value = 0.1576
## alternative hypothesis: stationaryTime Series 2: Temperature
store4_temp.ts<-ts(store4_data$Temperature,start =c(2010,5),freq=52)
tseda(store4_temp.ts,"Year","Temperature","Temperature by
Year") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -2.7561, Lag order = 5, p-value = 0.2617
## alternative hypothesis: stationaryTime Series 3: Unemployment Rate
store4_ue.ts<-ts(store4_data$Unemployment,start = c(2010,5),freq=52)
tseda(store4_ue.ts,"Year","Unemployment Rate","Unemployment Rate by Year") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -3.6827, Lag order = 5, p-value = 0.02829
## alternative hypothesis: stationaryTime Series 4: Fuel Price
store4_fp.ts<-ts(store4_data$Fuel_Price,start = c(2010,5),freq=52)
tseda(store4_fp.ts,"Year","Fuel Price","Fuel Price by Year") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -2.5205, Lag order = 5, p-value = 0.3597
## alternative hypothesis: stationaryTime Series 5: CPI
store4_cpi.ts<-ts(store4_data$CPI,start =c(2010,5),freq=52)
tseda(store4_sales.ts,"Year","CPI","CPI by Year") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -3.0062, Lag order = 5, p-value = 0.1576
## alternative hypothesis: stationaryConverting non-stationary time series to stationary series by differencing
Time Series 1: Sales
store4_sales.d.ts<-diff(store4_sales.ts,differences=1,lag=1)
tseda(store4_sales.d.ts,"Year","Sales at Store 4","Store 4 Sales by Year (Differenced)") 
## Warning in adf.test(timeseries): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -5.3952, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationaryBy differencing the “sales” time series once, we’re able to see via the plot and the ADF test that the differenced series is stationary. This means that the “sales” series was integrated with order 1 (I(1)). The differenced series has been stored in a new variable (store4_sales.d.ts)
Time Series 2: Temperature
store4_temp.d.ts<-diff(store4_temp.ts,differences=1,lag=1)
tseda(store4_temp.d.ts,"Year","Temperature","Temperature by Year (Differenced)")
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -2.9376, Lag order = 5, p-value = 0.1861
## alternative hypothesis: stationaryDifferencing the “temperature” time series once has created a time series that “looks” stationary with the first order difference. However, the ADT test says that the null hypothesis that the process is still unit root can’t be rejected, and this can be seen in the damping of the ACF and the pacfs which are slightly significant. However, we want to avoid over-differencing and for our purposes here, we’re dont think our analysis would be impacted if we assumed that temperature is an I(1) series.
Time Series 3: Unemployment
store4_ue.d.ts<- diff(store4_ue.ts,differences=1,lag=1)
tseda(store4_ue.d.ts,"Year","Unemployment Rate","Unemployment Rate by Year (Differenced)") 
## Warning in adf.test(timeseries): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -6.6314, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationaryAnalyzing the unemployment variable after the first order difference, the ADF function returns a p value of 0.01 rejecting the null hypothesis of unit root stationarity
Time Series 4: Fuel Price
store4_fp.d.ts<-diff(store4_fp.ts,differences=1,lag=1)
tseda(store4_fp.d.ts,"Year","Fuel Price","Fuel Price by Year (Differenced)") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -3.4417, Lag order = 5, p-value = 0.05046
## alternative hypothesis: stationaryThe first order difference of the “fuel price” time series satisfies the conditions of stationarity from an analysis of the acf,pacf and the ADF test.
Time Series 5: CPI
store4_cpi.d.ts<-diff(store4_cpi.ts,differences=1,lag=1)
tseda(store4_cpi.d.ts,"Year","CPI","CPI by Year (Differenced)") 
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -2.7002, Lag order = 5, p-value = 0.285
## alternative hypothesis: stationarystore4_cpi.d.ts<-diff(store4_cpi.d.ts,differences=1,lag=1)
tseda(store4_cpi.d.ts,"Year","CPI","CPI by Year (Second Order Differenced)") 
## Warning in adf.test(timeseries): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: timeseries
## Dickey-Fuller = -4.5331, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationaryAfter differencing once, the store4_cpi.ts time series continues to show strong evidence that it is a unit root series (through the ACF, PACF and ADF test). We have differenced the series once more to make it stationary.
Analyze cross-correlations
Ccf(store4_sales.d.ts,store4_temp.d.ts) 
Ccf(store4_sales.d.ts,store4_cpi.d.ts) 
Ccf(store4_sales.d.ts,store4_ue.d.ts) 
Ccf(store4_sales.d.ts,store4_fp.d.ts) 
Cross correlations are evaluated against differenced time series below.
- Based on the cross-correlation betwen temperature and sales, it appears as though there may not be a significant relationship between these two time series after the first 2-3 lags, although the first lag seems to be somewhat strongly correlated
*CPI and Sales do not seem to have any cross correlations of any importance at all
*Unemployment and Sales seem to have some small cross-correlations of interest although many seem coincidental (for example the 0.2 cross correlation at around lag 60)
*Fuel price and sales do not seem to have any cross correlations of any importance at all
We will model all these relationships during estimation, but based on these cross-correlations, the relationship between Sales and Temperature seems to be the most important, while unemployment may be of secondary importance.
Analyze cointegrations
po.test(cbind(store4_temp.ts,store4_sales.ts))##
## Phillips-Ouliaris Cointegration Test
##
## data: cbind(store4_temp.ts, store4_sales.ts)
## Phillips-Ouliaris demeaned = -22.979, Truncation lag parameter =
## 1, p-value = 0.03126po.test(cbind(diff(store4_cpi.ts),store4_sales.ts))##
## Phillips-Ouliaris Cointegration Test
##
## data: cbind(diff(store4_cpi.ts), store4_sales.ts)
## Phillips-Ouliaris demeaned = -15.12, Truncation lag parameter = 1,
## p-value = 0.1449po.test(cbind(store4_ue.ts,store4_sales.ts))## Warning in po.test(cbind(store4_ue.ts, store4_sales.ts)): p-value greater
## than printed p-value##
## Phillips-Ouliaris Cointegration Test
##
## data: cbind(store4_ue.ts, store4_sales.ts)
## Phillips-Ouliaris demeaned = -1.9539, Truncation lag parameter =
## 1, p-value = 0.15po.test(cbind(store4_fp.ts,store4_sales.ts))## Warning in po.test(cbind(store4_fp.ts, store4_sales.ts)): p-value greater
## than printed p-value##
## Phillips-Ouliaris Cointegration Test
##
## data: cbind(store4_fp.ts, store4_sales.ts)
## Phillips-Ouliaris demeaned = -2.7698, Truncation lag parameter =
## 1, p-value = 0.15- Based on this analysis, it appears as if sales are cointegrated with temperature (p-value of 0.03). In order to correctly model cointegrated series, we would need to use VECM (Vector Error Correcton Models) which have not been covered in this course so far. For the purpose of this lab, we will proceed by building standard VAR models after making a note of this detail.
Estimation
Break data into in-sample and out of Sample Data Sets
We are breaking the time-series’ up into in-sample and out-of-sample components. The in-sample set has 129 rows while the out of sample data contains 13 rows (10%).
istartindex=2
iendindex=130
ostartindex=iendindex+1
oendindex=dim(as.matrix(store4_sales.d.ts))[c(1)]
modelinput = cbind(store4_sales.d.ts,store4_temp.d.ts,store4_ue.d.ts,store4_fp.d.ts,store4_cpi.d.ts)[istartindex:iendindex,]
modeloos=cbind(store4_sales.d.ts,store4_temp.d.ts,store4_ue.d.ts,store4_fp.d.ts,store4_cpi.d.ts)[ostartindex:oendindex,]Estimate VAR model
This function will be used to calculate out of sample RSME’s for our models
oosrmse<-function(model){
p<-predict(model,n.ahead=oendindex-iendindex+1)
rmse(p$fcst$store4_sales.d.ts[,c(1)],modeloos[,c(1)])
}AR Model with Sales alone
modelar<-ar(modelinput[,c(1)],method = 'ols',dmean=T,intercept=F)
summary(modelar) ## Length Class Mode
## order 1 -none- numeric
## ar 3 -none- numeric
## var.pred 1 -none- numeric
## x.mean 1 -none- numeric
## x.intercept 0 -none- NULL
## aic 22 -none- numeric
## n.used 1 -none- numeric
## order.max 1 -none- numeric
## partialacf 0 -none- NULL
## resid 129 -none- numeric
## method 1 -none- character
## series 1 -none- character
## frequency 1 -none- numeric
## call 5 -none- call
## asy.se.coef 2 -none- listmodelar$ar## , , 1
##
## [,1]
## [1,] -0.4252121
## [2,] -0.2242020
## [3,] -0.1420759modelar$aic## 0 1 2 3 4 5
## 17.0175335 4.7734886 0.2639013 0.0000000 2.6019130 3.2694219
## 6 7 8 9 10 11
## 6.0285055 9.0581929 7.0816763 7.3688286 6.3327024 4.8976876
## 12 13 14 15 16 17
## 1.3652162 1.5722340 3.2096832 5.0598598 3.6140768 6.7368885
## 18 19 20 21
## 7.5433646 10.6961916 5.2510607 6.6547486For a simple AR model, the fit only considers the first three lag terms. This gives us a hint which is further validated by the Acf and Pacf of sales that higher order lag terms may not be significant when it pertains to the sales variable.
Model1: Sales+Temperature
VARselect(modelinput[,c(1,2)], lag.max = 6) ## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 5 1 1 5
##
## $criteria
## 1 2 3 4 5
## AIC(n) 1.698624e+01 1.699175e+01 1.700132e+01 1.698986e+01 1.695441e+01
## HQ(n) 1.704197e+01 1.708462e+01 1.713134e+01 1.715702e+01 1.715872e+01
## SC(n) 1.712342e+01 1.722038e+01 1.732140e+01 1.740140e+01 1.745740e+01
## FPE(n) 2.382544e+07 2.395859e+07 2.419277e+07 2.392370e+07 2.310047e+07
## 6
## AIC(n) 1.698216e+01
## HQ(n) 1.722362e+01
## SC(n) 1.757661e+01
## FPE(n) 2.376553e+07model1<-VAR(modelinput[,c(1,2)],p=1,ic="AIC")
summary(model1) ##
## VAR Estimation Results:
## =========================
## Endogenous variables: store4_sales.d.ts, store4_temp.d.ts
## Deterministic variables: const
## Sample size: 128
## Log Likelihood: -1447.808
## Roots of the characteristic polynomial:
## 0.3599 0.06589
## Call:
## VAR(y = modelinput[, c(1, 2)], p = 1, ic = "AIC")
##
##
## Estimation results for equation store4_sales.d.ts:
## ==================================================
## store4_sales.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.3916 0.0820 -4.776 4.92e-06 ***
## store4_temp.d.ts.l1 65.5567 18.4252 3.558 0.000529 ***
## const -3.9275 89.6887 -0.044 0.965141
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1011 on 125 degrees of freedom
## Multiple R-Squared: 0.1932, Adjusted R-squared: 0.1802
## F-statistic: 14.96 on 2 and 125 DF, p-value: 1.494e-06
##
##
## Estimation results for equation store4_temp.d.ts:
## =================================================
## store4_temp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.0001577 0.0004025 -0.392 0.696
## store4_temp.d.ts.l1 -0.0341590 0.0904294 -0.378 0.706
## const 0.3866036 0.4401846 0.878 0.381
##
##
## Residual standard error: 4.963 on 125 degrees of freedom
## Multiple R-Squared: 0.002943, Adjusted R-squared: -0.01301
## F-statistic: 0.1845 on 2 and 125 DF, p-value: 0.8317
##
##
##
## Covariance matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts
## store4_sales.d.ts 1022678 1090.69
## store4_temp.d.ts 1091 24.63
##
## Correlation matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts
## store4_sales.d.ts 1.0000 0.2173
## store4_temp.d.ts 0.2173 1.0000oosrmse(model1)## Warning in timeseries1 - timeseries2: longer object length is not a
## multiple of shorter object length## [1] 1055.817After adding temperature, VARSelect recommends an order of 5 based on AIC. However, there isn’t a significant difference between VAR(1) and VAR(5) in terms of AIC and so in the interest of parsimony, we are choosing a VAR(1) model
Model2: Sales+Temperature+Unemployment
VARselect(modelinput[,c(1,2,3)], lag.max = 6) ## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 6 1 1 6
##
## $criteria
## 1 2 3 4 5
## AIC(n) 12.98761 13.06180 13.05077 13.03633 13.01513
## HQ(n) 13.09905 13.25682 13.32938 13.39853 13.46091
## SC(n) 13.26197 13.54193 13.73666 13.92800 14.11257
## FPE(n) 436994.00327 470789.29483 465955.32818 459875.87305 451160.00615
## 6
## AIC(n) 12.92464
## HQ(n) 13.45400
## SC(n) 14.22785
## FPE(n) 413379.12443model2<-VAR(modelinput[,c(1,2,3)],p=1,ic="AIC")
summary(model2) ##
## VAR Estimation Results:
## =========================
## Endogenous variables: store4_sales.d.ts, store4_temp.d.ts, store4_ue.d.ts
## Deterministic variables: const
## Sample size: 128
## Log Likelihood: -1364.818
## Roots of the characteristic polynomial:
## 0.3528 0.09082 0.09082
## Call:
## VAR(y = modelinput[, c(1, 2, 3)], p = 1, ic = "AIC")
##
##
## Estimation results for equation store4_sales.d.ts:
## ==================================================
## store4_sales.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.38433 0.08089 -4.751 5.49e-06 ***
## store4_temp.d.ts.l1 63.92182 18.17471 3.517 0.00061 ***
## store4_ue.d.ts.l1 1466.65193 677.13678 2.166 0.03223 *
## const 48.72289 91.67446 0.531 0.59604
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 996.7 on 124 degrees of freedom
## Multiple R-Squared: 0.2226, Adjusted R-squared: 0.2038
## F-statistic: 11.83 on 3 and 124 DF, p-value: 7.204e-07
##
##
## Estimation results for equation store4_temp.d.ts:
## =================================================
## store4_temp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.0001606 0.0004044 -0.397 0.692
## store4_temp.d.ts.l1 -0.0335023 0.0908606 -0.369 0.713
## store4_ue.d.ts.l1 -0.5891476 3.3852013 -0.174 0.862
## const 0.3654542 0.4583070 0.797 0.427
##
##
## Residual standard error: 4.983 on 124 degrees of freedom
## Multiple R-Squared: 0.003187, Adjusted R-squared: -0.02093
## F-statistic: 0.1321 on 3 and 124 DF, p-value: 0.9408
##
##
## Estimation results for equation store4_ue.d.ts:
## ===============================================
## store4_ue.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -1.080e-06 1.067e-05 -0.101 0.91956
## store4_temp.d.ts.l1 2.381e-03 2.398e-03 0.993 0.32263
## store4_ue.d.ts.l1 -7.769e-02 8.933e-02 -0.870 0.38611
## const -3.923e-02 1.209e-02 -3.244 0.00152 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1315 on 124 degrees of freedom
## Multiple R-Squared: 0.01344, Adjusted R-squared: -0.01042
## F-statistic: 0.5633 on 3 and 124 DF, p-value: 0.6402
##
##
##
## Covariance matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 993343.158 1.115e+03 -6.11371
## store4_temp.d.ts 1114.587 2.483e+01 0.02263
## store4_ue.d.ts -6.114 2.263e-02 0.01729
##
## Correlation matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 1.00000 0.22444 -0.04666
## store4_temp.d.ts 0.22444 1.00000 0.03454
## store4_ue.d.ts -0.04666 0.03454 1.00000oosrmse(model2)## Warning in timeseries1 - timeseries2: longer object length is not a
## multiple of shorter object length## [1] 1058.585After adding Unemployment, VARSelect recommends an order of 6 based on AIC. However, there isn’t a significant difference between VAR(1) and VAR(6) in terms of AIC and so in the interest of parsimony, we are choosing a VAR(1) model
Model3: Sales+Temperature+Unemployment+FuelPrice
VARselect(modelinput[,c(1,2,3,4)], lag.max = 6) ## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 1 1 1 1
##
## $criteria
## 1 2 3 4 5
## AIC(n) 6.866215 6.959583 7.012296 7.019004 6.980806
## HQ(n) 7.051955 7.293915 7.495221 7.650521 7.760914
## SC(n) 7.323481 7.782661 8.201187 8.573708 8.901322
## FPE(n) 959.482893 1054.298616 1113.725653 1125.661705 1090.415310
## 6
## AIC(n) 6.968931
## HQ(n) 7.897632
## SC(n) 9.255260
## FPE(n) 1087.779988model3<-VAR(modelinput[,c(1,2,3,4)],p=1,ic="AIC")
summary(model3) ##
## VAR Estimation Results:
## =========================
## Endogenous variables: store4_sales.d.ts, store4_temp.d.ts, store4_ue.d.ts, store4_fp.d.ts
## Deterministic variables: const
## Sample size: 128
## Log Likelihood: -1145.954
## Roots of the characteristic polynomial:
## 0.5493 0.3504 0.07766 0.07766
## Call:
## VAR(y = modelinput[, c(1, 2, 3, 4)], p = 1, ic = "AIC")
##
##
## Estimation results for equation store4_sales.d.ts:
## ==================================================
## store4_sales.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.38477 0.08121 -4.738 5.86e-06 ***
## store4_temp.d.ts.l1 63.77225 18.25310 3.494 0.000663 ***
## store4_ue.d.ts.l1 1460.15047 680.18460 2.147 0.033780 *
## store4_fp.d.ts.l1 414.71646 1629.34060 0.255 0.799511
## const 46.02248 92.63171 0.497 0.620194
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1000 on 123 degrees of freedom
## Multiple R-Squared: 0.223, Adjusted R-squared: 0.1977
## F-statistic: 8.824 on 4 and 123 DF, p-value: 2.687e-06
##
##
## Estimation results for equation store4_temp.d.ts:
## =================================================
## store4_temp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.0001775 0.0003998 -0.444 0.6578
## store4_temp.d.ts.l1 -0.0392399 0.0898507 -0.437 0.6631
## store4_ue.d.ts.l1 -0.8385523 3.3482034 -0.250 0.8027
## store4_fp.d.ts.l1 15.9090916 8.0204165 1.984 0.0495 *
## const 0.2618627 0.4559789 0.574 0.5668
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 4.925 on 123 degrees of freedom
## Multiple R-Squared: 0.03408, Adjusted R-squared: 0.002673
## F-statistic: 1.085 on 4 and 123 DF, p-value: 0.3669
##
##
## Estimation results for equation store4_ue.d.ts:
## ===============================================
## store4_ue.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -1.140e-06 1.071e-05 -0.106 0.91546
## store4_temp.d.ts.l1 2.361e-03 2.408e-03 0.980 0.32884
## store4_ue.d.ts.l1 -7.858e-02 8.973e-02 -0.876 0.38289
## store4_fp.d.ts.l1 5.628e-02 2.149e-01 0.262 0.79388
## const -3.960e-02 1.222e-02 -3.240 0.00154 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.132 on 123 degrees of freedom
## Multiple R-Squared: 0.01399, Adjusted R-squared: -0.01807
## F-statistic: 0.4364 on 4 and 123 DF, p-value: 0.7821
##
##
## Estimation results for equation store4_fp.d.ts:
## ===============================================
## store4_fp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -3.666e-07 3.699e-06 -0.099 0.921
## store4_temp.d.ts.l1 4.874e-05 8.313e-04 0.059 0.953
## store4_ue.d.ts.l1 -5.071e-02 3.098e-02 -1.637 0.104
## store4_fp.d.ts.l1 5.583e-01 7.421e-02 7.523 9.75e-12 ***
## const 1.285e-03 4.219e-03 0.305 0.761
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.04556 on 123 degrees of freedom
## Multiple R-Squared: 0.3224, Adjusted R-squared: 0.3004
## F-statistic: 14.63 on 4 and 123 DF, p-value: 8.383e-10
##
##
##
## Covariance matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 1.001e+06 1103.42546 -6.2349501
## store4_temp.d.ts 1.103e+03 24.25254 0.0200676
## store4_ue.d.ts -6.235e+00 0.02007 0.0174172
## store4_fp.d.ts 2.077e+00 -0.01405 0.0001282
## store4_fp.d.ts
## store4_sales.d.ts 2.0770486
## store4_temp.d.ts -0.0140516
## store4_ue.d.ts 0.0001282
## store4_fp.d.ts 0.0020760
##
## Correlation matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 1.00000 0.22396 -0.04722
## store4_temp.d.ts 0.22396 1.00000 0.03088
## store4_ue.d.ts -0.04722 0.03088 1.00000
## store4_fp.d.ts 0.04557 -0.06262 0.02132
## store4_fp.d.ts
## store4_sales.d.ts 0.04557
## store4_temp.d.ts -0.06262
## store4_ue.d.ts 0.02132
## store4_fp.d.ts 1.00000oosrmse(model3)## Warning in timeseries1 - timeseries2: longer object length is not a
## multiple of shorter object length## [1] 1059.666After adding Fuel Price, VARSelect continues to recommend an order of 1. We use this order to build the model.
Model4: Sales+Temperature+Unemployment+FuelPrice+CPI
VARselect(modelinput[,c(1,2,3,4,5)], lag.max = 6) ## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 1 1 1 1
##
## $criteria
## 1 2 3 4 5 6
## AIC(n) -1.0024140 -0.9372290 -0.9238915 -0.8855573 -0.8265214 -0.7935867
## HQ(n) -0.7238037 -0.4264434 -0.1809307 0.0895787 0.3807899 0.6458999
## SC(n) -0.3165154 0.3202518 0.9051715 1.5150878 2.1457060 2.7502229
## FPE(n) 0.3671347 0.3926513 0.3999247 0.4195083 0.4519581 0.4780252model4<-VAR(modelinput[,c(1,2,3,4,5)],p=1,ic="AIC")
summary(model4) ##
## VAR Estimation Results:
## =========================
## Endogenous variables: store4_sales.d.ts, store4_temp.d.ts, store4_ue.d.ts, store4_fp.d.ts, store4_cpi.d.ts
## Deterministic variables: const
## Sample size: 128
## Log Likelihood: -811.817
## Roots of the characteristic polynomial:
## 0.5437 0.3523 0.244 0.07825 0.07825
## Call:
## VAR(y = modelinput[, c(1, 2, 3, 4, 5)], p = 1, ic = "AIC")
##
##
## Estimation results for equation store4_sales.d.ts:
## ==================================================
## store4_sales.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + store4_cpi.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -3.909e-01 8.177e-02 -4.781 4.92e-06 ***
## store4_temp.d.ts.l1 6.336e+01 1.829e+01 3.464 0.000736 ***
## store4_ue.d.ts.l1 1.458e+03 6.814e+02 2.140 0.034350 *
## store4_fp.d.ts.l1 3.290e+02 1.636e+03 0.201 0.840996
## store4_cpi.d.ts.l1 -3.586e+03 4.760e+03 -0.753 0.452722
## const 4.573e+01 9.280e+01 0.493 0.623062
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1002 on 122 degrees of freedom
## Multiple R-Squared: 0.2266, Adjusted R-squared: 0.1949
## F-statistic: 7.148 on 5 and 122 DF, p-value: 6.748e-06
##
##
## Estimation results for equation store4_temp.d.ts:
## =================================================
## store4_temp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + store4_cpi.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.0001760 0.0004034 -0.436 0.6634
## store4_temp.d.ts.l1 -0.0391399 0.0902580 -0.434 0.6653
## store4_ue.d.ts.l1 -0.8380628 3.3619045 -0.249 0.8036
## store4_fp.d.ts.l1 15.9299309 8.0727425 1.973 0.0507 .
## store4_cpi.d.ts.l1 0.8712465 23.4851864 0.037 0.9705
## const 0.2619346 0.4578453 0.572 0.5683
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 4.945 on 122 degrees of freedom
## Multiple R-Squared: 0.0341, Adjusted R-squared: -0.005491
## F-statistic: 0.8613 on 5 and 122 DF, p-value: 0.5093
##
##
## Estimation results for equation store4_ue.d.ts:
## ===============================================
## store4_ue.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + store4_cpi.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -1.349e-06 1.081e-05 -0.125 0.9009
## store4_temp.d.ts.l1 2.347e-03 2.418e-03 0.970 0.3338
## store4_ue.d.ts.l1 -7.864e-02 9.008e-02 -0.873 0.3844
## store4_fp.d.ts.l1 5.336e-02 2.163e-01 0.247 0.8056
## store4_cpi.d.ts.l1 -1.220e-01 6.293e-01 -0.194 0.8466
## const -3.961e-02 1.227e-02 -3.229 0.0016 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1325 on 122 degrees of freedom
## Multiple R-Squared: 0.0143, Adjusted R-squared: -0.0261
## F-statistic: 0.3539 on 5 and 122 DF, p-value: 0.8789
##
##
## Estimation results for equation store4_fp.d.ts:
## ===============================================
## store4_fp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + store4_cpi.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 6.115e-08 3.712e-06 0.016 0.987
## store4_temp.d.ts.l1 7.732e-05 8.306e-04 0.093 0.926
## store4_ue.d.ts.l1 -5.057e-02 3.094e-02 -1.635 0.105
## store4_fp.d.ts.l1 5.642e-01 7.429e-02 7.595 6.92e-12 ***
## store4_cpi.d.ts.l1 2.489e-01 2.161e-01 1.152 0.252
## const 1.306e-03 4.213e-03 0.310 0.757
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.0455 on 122 degrees of freedom
## Multiple R-Squared: 0.3297, Adjusted R-squared: 0.3022
## F-statistic: 12 on 5 and 122 DF, p-value: 1.857e-09
##
##
## Estimation results for equation store4_cpi.d.ts:
## ================================================
## store4_cpi.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + store4_fp.d.ts.l1 + store4_cpi.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 7.201e-07 1.511e-06 0.477 0.63453
## store4_temp.d.ts.l1 -1.639e-04 3.380e-04 -0.485 0.62871
## store4_ue.d.ts.l1 1.665e-03 1.259e-02 0.132 0.89500
## store4_fp.d.ts.l1 -1.425e-02 3.024e-02 -0.471 0.63837
## store4_cpi.d.ts.l1 2.349e-01 8.796e-02 2.670 0.00861 **
## const 1.769e-04 1.715e-03 0.103 0.91799
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.01852 on 122 degrees of freedom
## Multiple R-Squared: 0.06096, Adjusted R-squared: 0.02247
## F-statistic: 1.584 on 5 and 122 DF, p-value: 0.1696
##
##
##
## Covariance matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 1.004e+06 1.114e+03 -6.445e+00
## store4_temp.d.ts 1.114e+03 2.445e+01 2.027e-02
## store4_ue.d.ts -6.445e+00 2.027e-02 1.755e-02
## store4_fp.d.ts 2.418e+00 -1.425e-02 1.403e-04
## store4_cpi.d.ts -1.712e+00 -2.981e-03 2.307e-05
## store4_fp.d.ts store4_cpi.d.ts
## store4_sales.d.ts 2.418e+00 -1.712e+00
## store4_temp.d.ts -1.425e-02 -2.981e-03
## store4_ue.d.ts 1.403e-04 2.307e-05
## store4_fp.d.ts 2.071e-03 -7.138e-05
## store4_cpi.d.ts -7.138e-05 3.430e-04
##
## Correlation matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 1.00000 0.22471 -0.048536
## store4_temp.d.ts 0.22471 1.00000 0.030940
## store4_ue.d.ts -0.04854 0.03094 1.000000
## store4_fp.d.ts 0.05303 -0.06331 0.023266
## store4_cpi.d.ts -0.09224 -0.03256 0.009403
## store4_fp.d.ts store4_cpi.d.ts
## store4_sales.d.ts 0.05303 -0.092238
## store4_temp.d.ts -0.06331 -0.032557
## store4_ue.d.ts 0.02327 0.009403
## store4_fp.d.ts 1.00000 -0.084702
## store4_cpi.d.ts -0.08470 1.000000oosrmse(model4)## Warning in timeseries1 - timeseries2: longer object length is not a
## multiple of shorter object length## [1] 1059.279After adding CPI, VARSelect continues to recommend an order of 1. We use this order to build the model.
Model Selection
| Model | Time Series’ Included | Adjusted R2 | RMSE |
|---|---|---|---|
| Model1 | Sales+Temperature | 0.18 | 1055.817 |
| Model2 | Sales+Temperature+Unemployment | 0.20 | 1058.585 |
| Model3 | Sales+Temperature+Unemployment+FuelPrice | 0.19 | 1059.666 |
| Model4 | Sales+Temperature+Unemployment+FuelPrice+CPI | 0.19 | 1059.279 |
Based on the R2 values and the RMSE of the out-of-sample projections vs the actual values, we have chosen Model2 as our final VAR model and the Sales, Temperature and Unemployment as the variables to include in the model.
Final VAR model
finalmodel<-VAR(modelinput[,c(1,2,3)],p=1,ic="AIC")
summary(finalmodel) ##
## VAR Estimation Results:
## =========================
## Endogenous variables: store4_sales.d.ts, store4_temp.d.ts, store4_ue.d.ts
## Deterministic variables: const
## Sample size: 128
## Log Likelihood: -1364.818
## Roots of the characteristic polynomial:
## 0.3528 0.09082 0.09082
## Call:
## VAR(y = modelinput[, c(1, 2, 3)], p = 1, ic = "AIC")
##
##
## Estimation results for equation store4_sales.d.ts:
## ==================================================
## store4_sales.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.38433 0.08089 -4.751 5.49e-06 ***
## store4_temp.d.ts.l1 63.92182 18.17471 3.517 0.00061 ***
## store4_ue.d.ts.l1 1466.65193 677.13678 2.166 0.03223 *
## const 48.72289 91.67446 0.531 0.59604
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 996.7 on 124 degrees of freedom
## Multiple R-Squared: 0.2226, Adjusted R-squared: 0.2038
## F-statistic: 11.83 on 3 and 124 DF, p-value: 7.204e-07
##
##
## Estimation results for equation store4_temp.d.ts:
## =================================================
## store4_temp.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -0.0001606 0.0004044 -0.397 0.692
## store4_temp.d.ts.l1 -0.0335023 0.0908606 -0.369 0.713
## store4_ue.d.ts.l1 -0.5891476 3.3852013 -0.174 0.862
## const 0.3654542 0.4583070 0.797 0.427
##
##
## Residual standard error: 4.983 on 124 degrees of freedom
## Multiple R-Squared: 0.003187, Adjusted R-squared: -0.02093
## F-statistic: 0.1321 on 3 and 124 DF, p-value: 0.9408
##
##
## Estimation results for equation store4_ue.d.ts:
## ===============================================
## store4_ue.d.ts = store4_sales.d.ts.l1 + store4_temp.d.ts.l1 + store4_ue.d.ts.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## store4_sales.d.ts.l1 -1.080e-06 1.067e-05 -0.101 0.91956
## store4_temp.d.ts.l1 2.381e-03 2.398e-03 0.993 0.32263
## store4_ue.d.ts.l1 -7.769e-02 8.933e-02 -0.870 0.38611
## const -3.923e-02 1.209e-02 -3.244 0.00152 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1315 on 124 degrees of freedom
## Multiple R-Squared: 0.01344, Adjusted R-squared: -0.01042
## F-statistic: 0.5633 on 3 and 124 DF, p-value: 0.6402
##
##
##
## Covariance matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 993343.158 1.115e+03 -6.11371
## store4_temp.d.ts 1114.587 2.483e+01 0.02263
## store4_ue.d.ts -6.114 2.263e-02 0.01729
##
## Correlation matrix of residuals:
## store4_sales.d.ts store4_temp.d.ts store4_ue.d.ts
## store4_sales.d.ts 1.00000 0.22444 -0.04666
## store4_temp.d.ts 0.22444 1.00000 0.03454
## store4_ue.d.ts -0.04666 0.03454 1.00000oosrmse(finalmodel)## Warning in timeseries1 - timeseries2: longer object length is not a
## multiple of shorter object length## [1] 1058.585Key insights from this model are * The first lag of sales and temperature seem to be significant. This makes sense since these two variables are cointegrated * The first lag of unemployment seems to be slightly significant
Diagnostics
par(mar = rep(2, 4))
plot.ts(resid(finalmodel)[,c(1)],main="Plot of Residuals of Sales Time Series")
Box.test(resid(finalmodel)[,c(1)],type = c("Ljung-Box"))##
## Box-Ljung test
##
## data: resid(finalmodel)[, c(1)]
## X-squared = 1.1923, df = 1, p-value = 0.2749#Tests for normality
hist(resid(finalmodel)[,c(1)],main = "Histogram of Residuals of Final Model",xlab="")
qqnorm(resid(finalmodel)[,c(1)],main = "QQ Plot of Residuals of Final Model",xlab="")
In the plots and tests above, we test the residuals from the chosen model (for sales only)for-
- Independence
- Normality
The residuals appear to be independent and identically distributed (iid). Based on the Ljung-Box test, the null hypothesis that the residuals are iid cannot be rejected. The histogram of the residuals and the qqplot further validate the normality of the residuals
Model Performance Evaluation
par(mfrow=c(1,1))
plot.ts(modelinput[,c(1)],col="navy",lty=2,main="Original vs Estimated Series (First Difference)",ylab = "Original and Estimated Values",xlim = c(0,120),ylim=c(-3000,3000))
par(new=T)
plot.ts(fitted(finalmodel)[,c(1)],col="red",xlim = c(0,120),ylim=c(-3000,3000),xlab="",ylab="")
par(mfrow=c(1,1))
plot.ts(cumsum(c(store4_sales.ts[1],modelinput[,c(1)])),col="navy",lty=2,main="Original vs Estimated Series (Integrated)",ylab = "Original and Estimated Values",xlim = c(0,120),ylim=c(4000,13000))
par(new=T)
plot.ts(cumsum(c(store4_sales.ts[1],fitted(finalmodel)[,c(1)])),col="red",xlim = c(0,120),ylim=c(4000,13000),xlab="",ylab="")
Based on the graphs above, we can conclude that the model provides a reasonably good fit to the sales data
Hypothesis testing
- Is there a relationship between temperature and sales?
There appears to be a strong relationship between sales and the first lag of temperature. Make a note here however that these two time series are cointegrated.
- Is there a relationship between CPI and sales?
There doesn’t appear to be a relationship between these two variables
- Is there a relationship between fuel price and sales?
There doesn’t appear to be a relationship between these two variables
- Is there a relationship between unemployment and sales?
There appears to be a relationship sales and the first lag of unemployment
Forecasting
p<-predict(finalmodel,n.ahead=20)
fc<-cumsum(union(fitted(finalmodel)[,c(1)],p$fcst$store4_sales.d.ts[,c(1)]))
fc<-ts(fc)
plot(fc, main = "Sales Forecast", xlab = "Time",ylab="Sales")
fc[c(143:148)]## [1] 2439.253 2453.896 2468.539 2483.182 2497.825 2512.468We forecast that on an average, the store will sell the units projected above in department 12 for the next 5 time weeks.